Experimental Study on Influencing Factors of Motion Responses for Air-Floating Tetrapod Bucket Foundation

Air floating transport is one of the key construction technologies of bucket foundation. The influences of draft, water depth and bucket spacing on the motion response characteristics of tetrapod bucket foundation (TBF) during air-floating transportation were studied by models tests. The results showed that with the increase of draft, the natural periods of heave motion increased, while the maximum amplitudes of oscillating motion decreased. The maximum amplitudes of heave motion decreased while pitch motion increased with the increasing of water depth; further, the period range of oscillating amplitude close to the maximum amplitude was expanded due to shallow water effect. With increasing bucket spacing, the maximum amplitudes of heave motion first increase and then decreased, whereas the maximum amplitudes of pitch motion decreased. Therefore, the favorable air-floating transportation performance can be achieved by choosing a larger bucket spacing under the condition of meeting the design requirements and reducing the draft under shallower water.


Introduction
As a new type of foundation, air-floating structure has the advantages of self-floating transportation, negative pressure sinking and reusability. It is actively adopted as a foundation for marginal oilfield development platform, breakwater, floating airport, artificial island, etc. (Cheung et al., 2000;Bie et al., 2002;Ikoma et al., 2006;He et al., 2012;Ding et al., 2016). In addition, owing to the good geological adaptability, the high load capability and near-20%lower cost than other types of foundations, bucket foundation, has been applied to the development of offshore wind turbine. It is expected to become one of the main foundation types in a water depth range of 5−50 m (Byrne et al., 2002;Ding et al., 2015;Fu et al., 2014;Houlsby et al., 2005;Lars, 2012;Zhang et al., 2015). Furthermore, when the water depth exceeds 50 m, floating foundation is generally selected as the foundation type of offshore wind power (Zhang, 2018). Bucket foundation can be sunk into the seabed as either the foundation of mooring system or floating wind turbine. Compared with the traditional semi-submersible platform, on the one hand, the air-floating structure can provide the buoyancy required for floating through the pressure difference between the internal and external pressure to reduce the construction materials and the costs. On the other hand, the wave loads on structure are buffered owing to the compressibility of air-cushion inside. Air-floating structure was more stable and required less mooring loads (Qi, 1994(Qi, , 1998. However, owing to the bottom opening and the internal pressurized gas, the dynamic and motion response characteristics of the bucket foundation are different from those of the conventional floating body. Seidl (1980) introduced an air pocket factor to describe the relationship between the compressibility of the trapped air and the resulting hydro-static stiffness of the system. A few design concepts of the air-floating platform supported by a single large air cushion and several cushions were investigated by model tests. Cheung et al. (2000) used a boundary integral equation to model the dynamic response of a pneumatic floating platform; the air pocket factor proposed by Seidl (1980) was introduced to consider the effect of air cushion. The numerical simulation results were verified by model tests. The results showed that the dynamic response of the pneumatic platform can be quite sensitive to the air-pocket factor over a range of wave periods. The air and water columns also acted as cushions in reducing the hydrodynamic loads on the structure. A three-dimensional calculation method using linearized adiabatic law for the air pressure in the cushions to determine the air cushion supported floating body was proposed by Pinkster (1997). The numerical model was found to agree well with model tests on barge with one or two cushion compartments (Pinkster, 1997;Pinkster et al., 1998). Van Kessel et al. (2007a, 2007b and Van Kessel (2010) verified that the existence of air cushion significantly affected the stability and motion response and reduced the ship bending moment of the structure. Malenica and Zalar (2000) improved the method and analyzed the heave hydrodynamic coefficients of the air cushion supported structure. Using a boundary integral equation method, Guret and Hermans (2001) extended the works of Malenica and Zalar (2000) to investigate the transfer functions of heave motion and the interior vertical freesurface displacement of an air-cushion supported structure in regular waves. Furthermore, Malenica and Zalar (2001) presented the semi-analytical solution of heave motion for air cushion supported cylinder in finite water depth. Based on this method, Liu (2012) investigated the wave loads, the hydrodynamic coefficients and motion responses of large scale bucket foundation. Bie et al. (2002) developed an air floating reducing coefficient, accounting for the difference in the restoring force coefficient between the air floating structures and the conventional buoy and established the equation for calculation of the buoyancy of air-floating structure. Lee and Newman (2000) used the higher-order panel program HIPAN and model tests to study the wave response of very large floating structures (VLFS) with the characteristic length to wavelength ratio larger than 10. The elevation of the interface was represented by an appropriate set of Fourier generalized modes with unknown amplitudes. Zhang (2018) proposed the aerodynamic coefficients of air cushion with different shapes and investigated the heave performances of air-cushion supported platform. The numerical results were carried into wave analysis of Massachusetts Institute of Technology (WAMIT). The reliability of Lee and Newman's extended boundary integral equation was verified by experiments (Zhang, 2018;Lee and Newman, 2016). Thiagarajan (2009) presented a theoretical formulation to evaluate the restoring moment of structures supported by air cushions at zero forward speed. Further, a correction factor for the metacentric height incorporating the air cushion effects was proposed. Liu et al. (2019aLiu et al. ( , 2019b used the ideal air state equation considering the compression effect of the air cushion in the bucket foundation and developed the oscillation equations of the multibucket foundations. The response characteristics for tripod bucket foundations with different drafts, water depths and spacing were investigated by multi-operational structural engineering simulator (MOSES) and model tests.
It is not difficult to find that the research on the stability and seakeeping of air floating structures not only requires the theoretical analysis and model test, but also needs the data of actual construction and operation during air-floating transportation. Although the influences of internal and external factors on the air-floating structures were analyzed in the above literature, the research on the motion characteristics of tetrapod bucket foundation (TBF) is far from enough. In the present study, the motions of TBF were investigated with a series of experiments in wave flume by measuring the translational acceleration and the rotation angle. The displacement signal was obtained by the quadratic integration of the translational acceleration signal. The motion characteristics of TBF were illustrated in a parametric study. Successively detailed analysis and discussion of response amplitude operator (RAO) were carried out considering the influence of draft, water depth and bucket spacing.

Physical model
Figs. 1−3 show the detailed geometric dimensions of the prototypes of TBF and the corresponding test models made of stainless steel. As shown in Figs. 1a, 2a and 3a, the diameter D and the height H of each bucket foundation in prototypes are 20 m and 10 m, respectively. The plane layout of the four buckets, which are connected into one body by plates with sufficient rigidity, is in the form of square. The lengths S of connecting steel plate are 10 m (equal to 0.5D) for Prototype 1, 20 m (equal to 1.0D) for Prototype 2 and 30 m (equal to 1.5D) for Prototype 3, respectively. The similarity ratio between the model and the prototype meets the geometric similarity and Froude similarity, which is 1:100. The diameter and the height of each bucket foundation in models are 0.2 m and 0.1 m, respectively. The wall thickness of the side wall and the top lid are both 1.0 mm. The detailed structural parameters and mass characteristics of Prototype 1 and Model 1, Prototype 2 and Model 2, Prototype 3 and Model 3 are shown in Table 1, Table 2 and  Table 3, respectively, where H d , M s , and J x (J y ) represent the draft, mass, moment of inertia about the x(y) axis of the global coordinate system, respectively.

Experiment setup
The experiment was carried out in the wave flume of the Key Laboratory of National Defense Engineering in Logistical Engineering University. The geometries of the flume are 30 m in length, 1.0 m in width and 2.0 m in height, respectively. Regular waves are generated by a hydraulic push plate wave-maker at one end of the flume, and the wave crest line is paralleled to the direction of the wave making plate. To eliminate the influence of interference signal and obtain sufficient time for data acquisition, a wave absorbing device which can effectively prevent wave reflection is arranged at the other end of the flume. As shown in Fig. 4, the model is arranged at the position of 12.0 m away from the wave maker, and Cartesian coordinate system is established at the center of the structure on the still water surface. The x axis is paralleled to the propagation direction of the incident wave, the y axis is paralleled to the direction of the plate of wave-maker, and the z axis is vertical upward. During the test, the wave height was measured by a wave probe located between the structure and the wave-maker, and the model data was obtained by a gyroscope arranged at the top center of the structure.

Regular wave tests
In order to determine the amplitude frequency response characteristics of heave and pitch motion, regular wave tests in a certain range of periods at the same wave height are required. Since the similarity scale of the experiment is 1:100, the similarity ratio of wave height is 1:100, and the similarity ratio of wave period is 1:10. In the test, the wave height is 0.02 m and the wave frequency range is 0.5−1.5 s. In order to study the influences of water depth and draft on the motion responses, the water depth was 0.2 m, 0.3 m and 0.4 m, and draft was set to 0.05 m, 0.06 m and 0.07 m, respectively. The data of motion were obtained at 200 samples per second using a serial port assistant and post-processed using Fourier analysis to recover the amplitude and phase information using MATLAB software.

Effects of drafts
The RAOs of heave displacement for Model 1 with different drafts at 20.0 m water depth are shown in Fig. 5a. The maximum amplitudes of heave motion with a draft of 5.0 m, 6.0 m and 7.0 m occur near the period of 7 s, 7 s, and 8 s, respectively. With increasing draft, the resonant periods of heave motion increase, while the maximum amplitude decreases. The reason is that the increasing draft increases the added mass and reduces the wave force in heave direction, and the heave stiffness is close to that of rigid-bottom body due to the small scale of the model, which leads to the increase of resonant period and the decrease of amplitude. As the draft increases, the period range of oscillation amplitude close to maximum amplitude is expanded. There are some reasons contributing to this phenomenon. On one hand, the hydrodynamic interactions between buckets with small spacing has a significant effect on the wave force; on the other hand, the clearance between the bottoms of bucket and seabed becomes smaller with the increase of draft, and the gravity center sinks due to the blockage effect of ocean fluid, which is equivalent to adding an increment to the heave motion.
The RAOs of pitch angle for Model 1 with different drafts at 20.0 m water depth are shown in Fig. 5b. It is observed that the maximum amplitude of pitch angle 4.53° appears near 7 s at 5.0 m draft, 3.81° at 6.0 m draft near 7 s, and 3.70° at 7.0 m draft around 8 s, respectively. With the   increase of draft, the maximum amplitude of pitch angle decreases. The amplitude change of the draft increase from 5.0 m to 6.0 m is significantly larger than that from 6.0 m to 7.0 m.
The RAOs of heave displacement and pitch angle for Model 2 with different drafts at 20.0 m water depth are shown in Fig. 6. The maximum amplitudes of heave dis-    With the increase of draft, the maximum amplitude of heave displacement decreased, while the maximum amplitude of pitch angle first decreased and then slightly increased. The amplitude change of the draft increase from 5.0 m to 6.0 m is significantly larger than that from 6.0 m to 7.0 m. Compared with Model 1, an interesting phenomenon is that the period range of oscillation amplitude close to maximum amplitude is narrowed as the draft increases. The most likely reason is that the hydrodynamic interactions between buckets are weakened and the oscillation of wave forces is reduced at wider spacing. The RAOs of heave displacement for Model 3 with different drafts at 20.0 m water depth are shown in Fig. 7a. It can be seen that the maximum amplitude of heave displacement 3.78 m at 5.0 m draft, 6.59 m at 6.0 m draft and 5.18 m at 7.0 m draft occurred near the periods of 7 s, 8 s, and 7 s, respectively. Unlike Model 1 and Model 2, the maximum amplitude of heave displacement increases first and then decreases with the increase of draft. In most periods, the amplitude at 5.0 m draft is smaller than that of other drafts. The main reason is that bucket spacing of Model 3 is larger than that of Model 1 and Model 2, the effects of hydrodynamic interactions between buckets on wave forces reduce, while the shallow water effect on heave motion expands.
The RAOs of pitch angle for Model 3 with different drafts at 20.0 m water depth are shown in Fig. 7b. Similar to Model 1 and Model 2, the maximum amplitude of pitch angle, which occurs around 7 s, decreases with the increasing draft. The amplitude of pitch angle decreases at a low period and increases when the period is larger than 7 s with the increasing draft. The main reason is that the larger the draft, the narrower the clearance between the bottoms of bucket and the seabed, and the larger the velocity and the lower the pressure. Additional rocking motion of the structure is caused by the pressure change fore and aft.

Effects of water depths
The RAOs of heave displacement for Model 1 with different water depths at 6.0 m draft are shown in Fig. 8a. As is shown, the maximum and minimum amplitude of heave displacement at a water depth of 20 m appear at 6 s and 15 s, which are 7.88 m and 0.22 m, respectively; the values occur at 12 s and 13 s at a water depth of 30 m, which are 8.07 m and 0.39 m, respectively, and at 7 s and 10 s at a water depth of 40 m, which are 7.97 m and 0.38 m, respectively. As the water depth increases, the maximum amplitude and amplitude change of heave displacement show a trend of first increasing and then decreasing. In particular, the amplitudes at several periods are close to the maximum amplitude at a water depth of 30 m; however, the amplitude change of the heave displacement at a water depth of 40 m  is much smaller thanthose at other water depths. The reason is that during the air-floating transport from deep water to shallow water, the clearance between the bottom of the bucket and seabed becomes smaller. The gravity center sinks due to the blockage effect of ocean fluid, which is equivalent to adding an increment to heave displacement.
The RAOs of pitch angle for Model 1 with different water depths at 6.0 m draft are shown in Fig. 8b. The maximum and minimum amplitude of pitch angle appear at 7 s and 14 s at a water depth of 20 m, which are 3.81° and 0.87° respectively; the values occur at 7 s and 14 s at a water depth of 30 m, which are 4.11° and 1.12°, respectively, and at 8 s and 15 s at a water depth of 40 m, which are 4.80° and 1.02°, respectively. With the increase of water depth, the amplitude and amplitude change of pitch angle show a trend of increasing. Unlike Model 1, the amplitude change of the pitch angle at a water depth of 40 m is much larger than those at other water depths. However, the amplitudes at several periods are close to the maximum amplitude at a water depth of 20 m. Due to the shallow water effect, with the decrease of the water depth, the distance between the structure and the seabed decreases, resulting in higher relative velocity and lower bottom pressure, and the pressure changes between the fore and aft of the structure cause additional rocking motion.
The RAOs of heave displacement for Model 2 with different water depths at 6.0m draft are shown in Fig. 9a. As the water depth increases, the maximum amplitude of heave displacement shows a decreasing trend, while the heave motion of the structure slows down. In particular, the amplitude of heave displacement is stable at 1.0 m with a large range of periods at a water depth of 40 m. The maximum amplitude only decreases by 3.13% with water depth increasing from 20 m to 30 m, while sharply decreases by 66.4% with water depth increasing from 30 m to 40 m. On contrary to Model 1, the amplitude that is close to the maximum amplitude occurs near the resonant period at a water depth of 20 m.
The RAOs of pitch angle for Model 2 with different water depths at 6.0 m draft are shown in Fig. 9b. Similar to Model 1, as the water depth increases, the period of the maximum amplitude increases while the maximum amplitude of pitch angle first decreases and then increases. the amplitude at a water depth of 40 m is much larger than those at other water depths. Although the heave motion at a water depth of 30 m varies fiercely with the change of period, the pitch motion shows a stable trend.
The RAOs of heave displacement and pitch angle for Model 3 with different water depths at 6.0 m draft are shown in Fig. 10. Unfortunately, data at the water depth of 40 m were lost due to improper operation during the tests. Therefore, data in this case should be made reasonable predictions to get some useful analysis. The RAOs of heave displacement and pitch angle for Model 1, Model 2, and  LIU Xian-qing et al. China Ocean Eng., 2022, Vol. 36, No. 2, P. 258-267 Model 3 with different water depths at 6.0 m draft are shown in Figs. 11a−11f. It can be seen from Fig. 11a and Fig. 11b, the maximum amplitude of heave displacement for Model 3 is smaller than that for Model 1 and Model 2, and the shallow water effect is significantly weakened with the increasing water depth. It can be inferred that the interac-tions between wave and structure are the main factors for heave motion of Model 3. Thus, the maximum amplitude and amplitude changes of the heave displacement for Model 3 at the water depth of 40 m are similar to those at the water depth 30 m. The maximum amplitude of heave displacement for Model 3 decreases with the increase of water
As shown in Fig. 11d, the maximum amplitude and amplitude change of pitch angle of Model 3 are samller than those of Model 1 and Model 2 at 20 m water depth. In Fig. 11e, these values of Model 1 are larger than those of Model 2 and Model 3 at 30 m water depth, but the differences of maximum amplitude and amplitude change between Model 2 and Model 3 are only 0.01° and 0.17°, respectively. Comparing Fig. 11d with Fig. 11e, it can be seen that the variation trends of Model 3 at water depth of 20 m and 30 m are similar, and the maximum amplitude increases by 0.24°. Thus, the maximum amplitude and amplitude changes of the pitch angle for Model 3 at the water depth of 40 m are similar to those at water depth of 20 m and 30 m. The maximum amplitude of pitch angle for Model 3 increases with the increase of water depth.

Effects of spacing
The RAOs of heave displacement and pitch angle with different spacing at 5.0 m draft at a water depth of 20 m are shown in Fig. 12. It can be seen from Fig. 12a that the maximum amplitudes of heave displacement for S=0.5D, S=1.0D and S=1.5D all occur near the period of 7 s, respectively, which are 13.25 m, 15.92 m and 3.78 m. As the spacing increases, the maximum amplitude first increases and then decreases sharply. As shown in Fig. 12b, the maximum amplitudes of pitching angle are 4.53° for S=0.5D, 3.93°f or S=1.0D, and 2.42° for S=1.5D, respectively. The pitch RAOs for S=1.5D were smaller than those for S=0.5D and S=1.0D at almost each period. The reason is that the pitch angle is inversely proportional to pitch stiffness, i.e. the wider the spacing, the large the stiffness and the smaller the angle.
The RAOs of heave displacement and pitch angle with different spacing at 6.0 m draft at a water depth of 20 m are shown in Figs. 11a and 11b. As shown in Fig 11a, the maximum amplitudes of heaving displacement for S=0.5D, S=1.0D and S=1.5D occur around 7 s, 7 s and 8 s, respectively, which are 7.86 m, 10.23 m and 6.59 m. Similar to the 5.0 m draft, the maximum amplitude first increases and then decreases with the increase of spacing. It can be seen from Fig. 11b that the RAOs for pitch angle decrease when the spacing increases. The periods for the maximum amplitude of pitch angle are all around 7 s. Comparing Fig. 12a with Fig. 12b, it is not difficult to find that the pitch motion can be significantly reduced with the increase of spacing, but the heave motion is enhanced at 1.0D spacing. Furthermore, the increase of spacing causes an increase in the amount of connecting materials and additional costs, which does not meet the requirements of low-cost for air-floating bucket foundation.
The RAOs of heave displacement and pitch angle with different spacing at 7.0 m draft at a water depth of 20 m are shown in Fig. 13. Comparing Fig. 11a, Fig. 11b with  LIU Xian-qing et al. China Ocean Eng., 2022, Vol. 36, No. 2, P. 258-267 Fig. 12, the maximum amplitudes of heave displacement and pitch angle show a similar trend with the increase of bucket spacing. The narrower the spacing, the wider the period range of oscillation amplitude closes to maximum amplitude. An interesting phenomenon is that the amplitudes of pitch motion in the period range of 13 s to 15 s increases with the increase of spacing. The main reason is that the motion responses, on the one hand are significantly affected by shallow water effect, and on the other hand affected by the ratio of characteristic length to wavelength. When the ratio is about 2.0, the structure fore and aft is located at the position of wave peak and trough, which enhances the vertical motion. The wavelength in the period range of 7-10 s is 72-121 m, which is about twice the size of the wavelength in Model 1 and Model 2; while the wavelength is 167-197 m in the period range of 13-15 s which is about twice the size of that in Model 3.

Conclusions
In this paper, the parameters of heave and pitch motion for TBF under regular waves are measured through a series of model tests. The effects of draft, water depth and spacing on the motion response characteristics are studied. The main conclusions are as follows: (1) Draft has an effect on the oscillating motion of the air-floating structure. With the increase of draft, the natural period and maximum amplitude of heave motion show an increasing trend, while the maximum amplitude of pitch motion decreases. However, the maximum amplitude of heave motion decreases with smaller spacing.
(2) Water depth is also the main factor affecting the heave and pitch motions. With the increase of water depth, the maximum amplitude and amplitude change of heave motion decrease, while the maximum amplitude of pitch motion increases. However, several amplitudes close to the maximum amplitude appear in the measured period at smaller water depth, the vertical movements of the structure under shallow water are enhanced.
(3) With the increase of bucket spacing, the maximum amplitude of heave motion shows a trend of increasing first and then decreasing, while the maximum amplitude of pitch motion shows a downward trend. The narrower the spacing, the wider the period range of oscillation amplitude closes to maximum amplitude. Although the increase of spacing causes an increase in the amount of connecting materials and additional costs, which does not meet the requirements of low-cost for air-floating bucket foundation. Therefore, on the premise of meeting design requirements, a reasonable spacing can be chosen for the foundation for offshore wind turbine to obtain the optimal transportation.